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On geometric invariants, learning, and recognition of shapes and forms. (English) Zbl 07225685

Grohs, Philipp (ed.) et al., Handbook of variational methods for nonlinear geometric data. Cham: Springer. 443-461 (2020).
Summary: Extracting meaningful representations from geometric data has prime importance in the areas of computer vision, computer graphics, and image processing. Classical approaches use tools from differential geometry for modeling the problem and employed efficient and robust numerical techniques to engineer them for a particular application. Recent advances in learning methods, particularly in the areas of deep-learning and neural networks provide an alternative mechanism of extracting meaningful features and doing data-engineering. These techniques have proven very successful for various kinds of visual and semantic cognition tasks achieving state-of-the art results. In this chapter we explore the synergy between these two seemingly disparate computational methodologies. First, we provide a short treatise on geometric invariants of planar curves and a scheme to discover them from data in a learning framework, where the invariants are modelled using neural networks. Secondly, we also demonstrate the reverse, that is, imputing principled geometric invariants like geometric moments into standard learning architectures enables a significant boost in performance. Our goal would not only be to achieve better performance, but also to provide a geometric insight into the learning process thereby establishing strong links between the two fields.
For the entire collection see [Zbl 07115003].

MSC:

65Dxx Numerical approximation and computational geometry (primarily algorithms)
65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65Nxx Numerical methods for partial differential equations, boundary value problems
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